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Cayley diagram

C3⋊#401 #4662

order192
isoclinismisoclinic class of #4656
automorphismAut(G) order 9216
homologyAbelianization (H1) = C23 · Schur multiplier (H2) = C22

Aliases

unambiguous semidirect productC3⋊#401

Splittings

SylowC3 ⋊ #401

Subgroups

subgroupG/H
center, FrattiniC23C22×D3
commutator, socle, prime radical, cosocleC23×C3C23
FittingC3×#95C2
hypercenter#95D3
hypocenterC3#401
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #95

118 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 27
transitive degreeμt(G) = 192

A minimal faithful permutation representation — 3 generators on 27 points:

(1,2,3)(5,9)(8,11)(14,19)(16,18)(20,22,23,27)(21,24,25,26);(4,6,7,10)(5,8,9,11)(12,14)(13,16)(15,19)(17,18)(22,27)(24,26);(2,3)(4,5)(6,8)(7,9)(10,11)(12,13,15,17)(14,18,19,16)(20,21,23,25)(22,26,27,24)

Polycyclic presentation:

2^6,3|a^2=f,b^2=d,c^2=ef,b^a=be,c^a=cdef,g^a=g^2,c^b=cf

Conjugacy classes & character table

42 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×C3⋊#401, #73353, #73356, …
maximal normal subgroup ofC2×C3⋊#401, #73353, #73356, …
maximal quotient ofC3⋊#2104, #69416, #69417, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(192, 221);